{
  "id": "2406.06065",
  "version": "v1",
  "published": "2024-06-10T07:22:14.000Z",
  "updated": "2024-06-10T07:22:14.000Z",
  "title": "Impossibility of decoding a translation invariant measure from a single set of positive Lebesgue measure",
  "authors": [
    "Aleksandar Bulj"
  ],
  "comment": "5 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $\\mu$ be a translation invariant measure on $(\\mathbb{R}^d,\\mathcal{B}(\\mathbb{R}^d))$ and let $\\lambda$ denote the Lebesgue measure on $\\mathbb{R}^d$. If there exists an open set $U$ such that $0<\\mu(U)=\\lambda(U)<\\infty$, it is a simple exercise to show that $\\mu=\\lambda|_{\\mathcal{B}(\\mathbb{R}^d)}$. Is the same conclusion true if $U$ is merely a Borel set? The main purpose of this short note is to construct a measure that provides a negative answer to this question. Incidentally, this construction provides a new example of a translation invariant measure with a rich domain and range that is not Hausdorff, a problem previously studied by Hirst. Although the constructed measure may exist in the literature, we were unable to locate any references to it.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-10T07:22:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "translation invariant measure",
      "positive lebesgue measure",
      "single set",
      "impossibility",
      "open set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}